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with dimension L2/T and u- nit St (abbreviation of Stokes) = 1 cmz/s in the CGS system or m2/s( = l o 4 St) in the SI system. h is a parameter related to the inflation of volume. As stated before, the Stokes hypothesis states that the trace of the tensor z equals zero, so from Eq. (1. 2. 39), h and ,u must satisfy the condition 3h+2,u=O. The quantity I+2,u/3 is called the sec- ond viscosity, while p is the first viscosity. For a Stokes' fluid, the second viscosity is equal to zero, A= - 2p/3 (cf. Eq. ( 1. 2. 4 1 ) ) . Stokes' hypothesis is accurate e- nough in engineering applications, at least for gases and Newtonian fluids. For a Stokes' fluid, it can be proved that only the viscosity normal stress exists when inflation rate in the same direction is not equal to the mean value over those in the three coordinate directions, while viscosity shear stress is proportional to shear de- formation rate. Especially in the case of liquids, there are two sources of viscosity shear stress. One is the momentum exchange and viscosity dissipation stemming from random thermal motions of molecules, but its magnitude is relatively small. The oth- er is the continuous redistribution of molecules due to the velocity gradient, incurring a rotation of the net force field among contiguous molecules. k-,uC,,(or ,uCJ , D ~ P 1111. NAVIER-STOKES ( N S ) EQUATIONS 1. NS equation in vector form Introducing the constitutive equations for isotropic Newtonian Stokes fluid (i. e. , under the assumptions that ,u=const, 3I+2,u=0) , Eqs. (1. 2. 28) and (1. 2. 4 1 ) , into the equation of motion, Eq. (1. 2. 25 ) , we obtain the NS equation where V ( V V ) is a divergence of the gradient of vector V . In a rectangular coor- dinate system, the operator is called the Laplace operator (Laplacian) , also denoted by V *(or A ) , since it can be expanded by taking a formal operation like scalar prod- uct 14 (1. 2. 45) In this case, the components of V ( V V ) can be obtained by applying V to the correspondingcomponentsof V , i .e. , V (V V ) = V 2 V = V VV (SOV V is called a quasi-vector) . However, this relation does not hold in a more general coordi- nate system, when the operator should be expanded into (1. 2. 46) For an incompressible fluid, the last term on the right-hand side of Eq. (1. 2. 44) e- quals zero. If we have also an ideal fluid ( p = 0) , a reduced form of the NS equa- tions is called Euler equations. But even if p# 0, when all the derivatives &,/ax, are sufficiently small, the flow can still be approximated by the Euler equations. When DV/Dt= 0, the NS equations are reduced to a linear system, Stokes equations. The associated flow, Stokes flow (not to be confused with the Stokes fluid) , occurs when the inertial force is much smaller than the viscous force. We note in passing that historically only the equations of motion were called the NS equation. Now this term often means the complete system, including the equation of continuity (and additionally, the equation of energy for a general compressible fluid). There are important differences in structure and solution between the NS equa- tions for compressible and incompressible fluids. For incompressible fluids, only ve- locity appears in the equation of continuity, containing no density and pressure. This equation, as it is not in a complete form, is only partly coupled with the equation of motion and thus can be considered as a constraint on the velocity field. Pressure is in an unequal position to velocity, and cannot be determined by some thermodynamic condition (e. g. , equation of state). In numerical solutions, pressure and velocity are often obtained alternately in an iterative process. A difficulty lies in that the ve- locity obtained from the equation of motion under a given pressure generally cannot fulfill the equation of continuity. As for a compressible fluid, density appears in the equation of continuity, which is fully coupled with the equation of motion. If we take a fixed region in a flow, the pressure gradient at its boundary determines a velocity of the fluid leaving that re- gion, based on the equation of motion. That velocity incurs conversely a variation of density in the region based on the equation of continuity, and this has a feedback ef- fect on the pressure through the equation of state. In numerical solutions, pressure and velocity are often solved for simultaneously, and the 'elastic' constraint provided by the equation of continuity make the procedure easier due to inter-adjustability be- tween pressure and velocity. 2. NS equations in rectangular coordinate system for incompressible flow Later in this section we discuss incompressible flow only, but thereafter we shall study 2-D shallow-water flow from the viewpoint of compressible flow. The equation of continuity is (1. 2. 47) 15 Only the equation of motion in the x1 direction is listed below For the sake of brevity, it may be expressed by the use of a Cartesian tensor Eqs. (1. 2. 47) and (1. 2. 48) may be written concisely as (1. 2. 49) (1. 2. 5 0 ) (1. 2. 51) (1. 2. 52) We note in passing that space coordinates and velocity components will occasion- ally be denoted by (xl , x 2 , x3) and (u l , u 2 , us> , and sometimes by ( x , y , z ) and ( u , v , w). The latter are chiefly used in the 2-D case. 3. The NS equations expressed in terms of the stream function for 2-D incompre sible inviscid irrotational flows In the 2-D cases (plane or axis-symmetry) , it is possible to eliminate the pres- sure p from Eq. (1. 2. 52). Moreover, since for incompressible flows, p does not ap- pear in the equation of continuity, u and v can be combined into one unknown func- tion, so that only one equation is left. Defining the stream function @ by (1.2. 53) and introducing the above equation into the NS equations with p= 0 , we obtain (1. 2 .54) (1.2. 55) Taking derivatives of the above two equations with respect to y and x respective- ly , and then eliminating p , we have (1. 2. 56) When the flow is vortex-free, i. e. , the vorticity equals zero everywhere, the condition V X V = 0 can be expanded in the 2-D case, yielding 16 = o a u a v --- ag ax (1.2. 57) Introducing Eq. (1. 2. 57) into Eq. (1. 2. 53) , we get a Laplace equation satis- fied by the stream function v2*= 0 (1. 2. 58) Such an irrotational flow is called a potential flow. Substitute Eq. (1. 2. 58) into Eq. (1. 2. 561, and solve for the stream function under a given boundary condition. Then by taking a derivative in Eq. (1. 2. 53) we get a solenoidal velocity field also satisfying the NS equations. Lastly, substitute the velocity into the NS equations to obtain the pressure gradient. The whole problem has now been solved. An advantage of this approach is that the difficulty stemming from nonlinear convective terms can be avoided. In an extension to the 3-D case, a scalar velocity potential function can be in- troduced to replace the three velocity components. For a steady, irrotational and isentropic flow with pressure as the only external force, the NS equations can again be reduced to a single PDE in terms of the velocity potential. 4. Simplification of the equation of motion for incompressible flows For a general incompressible flow, the stream function cannot be used advan- tangeously, so it is necessary to adopt another approach in order to cancel out the e- quation of continuity and so eliminate pressure from the equation of motion. Firstly, we introduce a theorem : Any vector field w defined on a region can be uniquely decomposed into w=V+ V p , where vector field V satisfies: (i) div V= 0; (ii) V n= 0 , where n is an outward normal vector to the boundary of Sa. Physical- ly, it expresses the continuity requirement and the boundary condition that vector Ir is in its tangential direction. Based on the theorem, we define a linear